A274885 Coefficients of some q-polynomials, P_n(q) = q_factorial(n+1) / (q_factorial([n/2]) * q_factorial([(n+2)/2])) with [.] the floor function.

1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1

Offset: 0

Author

Peter Luschny, Jul 20 2016

Keywords

• nonn
• tabf

Examples

			The polynomials start:
[0] 1
[1] q + 1
[2] q^2 + q + 1
[3] (q + 1) * (q^2 + 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
[5] (q + 1)*(q^2 - q + 1)*(q^2 + 1)*(q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 1, 1]
[3] [1, 2, 3, 3, 2, 1]
[4] [1, 1, 2, 2, 2, 1, 1]
[5] [1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1]
[6] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1]
[7] [1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1]
		

Links

• G. C. Greubel, Rows n = 0..35 of triangle, flattened
• Peter Luschny, Orbitals

Crossrefs

Cf. Row sums are A212303(n+1) and A275212(n,0), A274886.

Programs

Magma

QFac:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
P:= func< n,x | QFac(n+1,x)/( QFac(Floor(n/2),x)*QFac(Floor((n+2)/2),x) ) >;
R:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( P(n,x) )): n in [0..8]]; // G. C. Greubel, May 22 2019

Maple

Qbinom1 := proc(n) local F, h; h := iquo(n,2);
F := x -> QDifferenceEquations:-QFactorial(x,q);
F(n+1)/(F(h)*F(h+1)); expand(simplify(expand(%)));
seq(coeff(%,q,j), j=0..degree(%)) end: seq(Qbinom1(n), n=0..8);

Mathematica

QBinom1[n_] := QFactorial[n+1,q] / (QFactorial[Quotient[n,2],q] QFactorial[Quotient[n+2,2],q]); Table[CoefficientList[QBinom1[n] // FunctionExpand,q], {n,0,8}] // Flatten

Sage

from sage.combinat.q_analogues import q_factorial
def q_binom1(n): return (q_factorial(n+1)//(q_factorial(n//2)* q_factorial((n+2)//2)))
for n in (0..10): print(q_binom1(n).list())